Difference Equation Explained by Oppenheim

[wildman]

This is some math from "Discrete-Time Signal Processing" by Oppenheim:

We have the homogeneous difference equation:

$$\sum_{k=0}^N a_k y_h [n-k] = 0$$

"The sequence $$y_h[n]$$ is in fact a member of a family of solutions of the form:

$$y_h[n] = \sum_{m=1}^ N A_m z^n_m$$"

So what is Oppenheim saying here? I suppose it is something like the solutions to differential equations. Right? But what is the z? I suppose it is related to the z transform, right?

He then says: Substituting the second equation for the first shows that the complex numbers $$z_m$$ must be roots of the polynomial:

$$\sum_{k=0}^N a_k z^{-k} = 0$$

Could some one explain this to me?

Thanks!

 

[ice109]

i think this might be what you're looking for
http://tutorial.math.lamar.edu/AllBrowsers/3401/SeriesSolutions.asp